Research On The Optimal Strategy Of Life Insurance Including Default Risk

With the prosperity and development of our country's economy,more and more families begin to consider buying insurance.Insurance not only can protect the risk of unexpected loss of family property,but also has the function of saving and investing.With the rapid development of economy,more and more investment instruments in financial market will be affected by default risk,such as default risks bonds and various credit derivatives.This paper assumes that investors can invest in bank deposits,stocks and bonds with default risk.Based on the theory of dynamic continuous time finance,this paper studies the optimal investment and consumption strategies of households under fixed and stochastic interest rates respectively.Utility function maximization is used to construct the optimal investment and consumption strategies.The model is established and solved by stochastic control and dynamic programming equation.The analytical solution and optimal strategy are obtained.Finally,the influence of household consumption weight on optimal strategy is discussed by numerical analysis.The following research has been done in this paper:Firstly,the life insurance problem with default risk under fixed interest rate is studied.Under the framework of simplified model,assuming that the bonds purchased by investors have default risk,this chapter studies the optimal consumption of households and the purchase of life insurance.Suppose that investors can deposit in investment banks,stocks and bonds with default risk,and use indicative function to violate default risks bonds.The contract risk is characterized and the HJB equation is deduced.The stochastic control method is used to solve the equation,and the corresponding analytical solution and optimal strategy are obtained when default risks bonds default or not.The results show that with the introduction of jump risk,the investor's investment strategy is no longer a continuous function of t at the investment time.Finally,the numerical analysis is used to discuss the investment strategy.The influence of household consumption weight on optimal strategy.Secondly,we study life insurance with default risk under stochastic interest rates.This chapter assumes that there are three parts of assets available for investment in the market: bank deposits,risky assets and bonds with default risk.Because life insurance contracts often last for a long time,investors need to consider interest rate risk.Therefore,this chapter assumes that interest rates and credit spreads are subject to CIR.The model guarantees that both interest rate and credit spreads are positive.Because investors will get work income,the model assumes that the work income of investors is fixed.Utility function maximization is used to establish the model,and dynamic equation method is used to solve the model,so the analytical solution and optimal strategy of the equation are obtained.Finally,we study the life insurance problem with default risk in the case of stochastic interest rate and stochastic wage.This chapter establishes a model under the utility of CRRA and assumes that the financial market is composed of bank deposits,stocks and bonds with default risk.We analyze the relationship among household consumption,insurance purchase and financial asset allocation.Investors need to consider interest rate risk,so this chapter uses the V asicek model to characterize interest rates and credit spreads.Because insured people will have work income,and income will change with the fluctuation of financial markets.Therefore,we use the Black-Scholes model to characterize investors' random income,and assume that there is a co-integration relationship between income growth and interest rates.When solving the HJB equation,we will use the non-homogeneous differential equation.The equation is transformed into homogeneous differential equation.The optimal solution of the model is obtained by using the method of dynamic programming principle,and the optimal strategy of the equation's analytic solution and wealth utility maximization is obtained.The influence of household consumption weight on the optimal strategy is discussed by numerical analysis.